In preceding chapters, we have discussed convective heat transfer with very important assumption. We have assumed a single-phase convective heat transfer without any phase change. In this chapter we focus on convective heat transfer associated with the change in phase of a fluid. In particular, we consider processes that can occur at a solid–liquid or solid–vapor interface, namely, boiling (liquid-to-vapor phase change).
For these cases latent heat effects associated with the phase change are significant. Latent heat, known also as the enthalpy of vaporization, is the amount of heat added to or removed from a substance to produce a change in phase. This energy breaks down the intermolecular attractive forces, and also must provide the energy necessary to expand the gas (the pΔV work). When latent heat is added, no temperature change occurs.
Boiling heat transfer is heat transferred by the boiling of water. In BWRs boiling of coolant occurs at normal operation and it is very desired phenomenon. During an unusual, transient, large amounts of additional core boiling could occur and safety limits could be exceeded. Therefore, boiling heat transfer is an extremely important topic for accident considerations, for both BWRs and PWRs. Boiling heat transfer is not a fourth method of heat transfer, but it is considered separately because of its importance.
Categorization by the wall superheat temperature, ΔTsat:
The pioneering work on boiling was done in 1934 by S. Nukiyama, who used electrically heated nichrome and platinum wires immersed in liquids in his experiments. Nukiyama was the first to identify different regimes of pool boiling using his apparatus. He noticed that boiling takes different forms, depending on the value of the wall superheat temperature ΔTsat (known also as the excess temperature), which is defined as the difference between the wall temperature, Twall and the saturation temperature, Tsat.
Four different boiling regimes of pool boiling (based on the excess temperature) are observed:
- Natural Convection Boiling ΔTsat < 5°C
- Nucleate Boiling 5°C < ΔTsat < 30°C
- Transition Boiling 30°C < ΔTsat < 200°C
- Film Boiling 200°C < ΔTsat
The most common type of local boiling encountered in nuclear facilities is nucleate boiling. But in case of nuclear reactors, nucleate boiling occurs at significant flow rates through the reactor. In nucleate boiling, steam bubbles form at the heat transfer surface and then break away and are carried into the main stream of the fluid. Such movement enhances heat transfer because the heat generated at the surface is carried directly into the fluid stream. Once in the main fluid stream, the bubbles collapse because the bulk temperature of the fluid is not as high as the heat transfer surface temperature where the bubbles were created. As was written, nucleate boiling at the surface effectively disrupts this stagnant layer and therefore nucleate boiling significantly improves the ability of a surface to transfer thermal energy to bulk fluid. This heat transfer process is sometimes desirable because the energy created at the heat transfer surface is quickly and efficiently “carried” away.
Close to the wall the situation is complex for several mechanisms increase the heat flux above that for pure conduction through the liquid.
- Note that, even in turbulent flow, there is a stagnant fluid film layer (laminar sublayer), that isolates the surface of the heat exchanger. The upward flux (due to buoyant forces) of vapor away from the wall must be balanced by an equal mass flux of liquid and this brings cooler liquid into closer proximity to the wall.
- The formation and movement of the bubbles turbulises the liquid near the wall and thus increases heat transfer from the wall to the liquid.
- Boiling differ from other forms of convection in that it depends on the latent heat of vaporization, which is very high for common pressures, therefore large amounts of heat can be transferred during boiling essentially at constant temperature.
The nucleate boiling heat flux cannot be increased indefinitely. At some value, we call it the “critical heat flux” (CHF), the steam produced can form an insulating layer over the surface, which in turn deteriorates the heat transfer coefficient. This is because a large fraction of the surface is covered by a vapor film, which acts as an thermal insulation due to the low thermal conductivity of the vapor relative to that of the liquid. Immediately after the critical heat flux has been reached, boiling become unstable and transition boiling occurs. The transition from nucleate boiling to film boiling is known as the “boiling crisis”. Since beyond the CHF point the heat transfer coefficient decreases, the transition to film boiling is usually inevitable.
In following section, we will distinguish between:
- nucleate pool boiling
- nucleate flow boiling
Nucleate Boiling Correlations – Pool Boiling
Boiling regimes discussed above differ considerably in their character. There are also different correlations that describe the heat transfer. In this section we review some of the more widely used correlations for nucleate boiling.
Nucleate Pool Boiling
The most widely used correlation for the rate of heat transfer in the nucleate pool boiling was proposed in 1952 by Rohsenow:
- q – nucleate pool boiling heat flux [W/m2]
- c1 — specific heat of liquid J/kg K
- ΔT — excess temperature °C or K
- hfg – enthalpy of vaporization, J/kg
- Pr — Prandtl number of liquid
- n — experimental constant equal to 1 for water and 1.7 for other fluids
- Csf — surface fluid factor, for example, water and nickel have a Csf of 0.006
- μ1 — dynamic viscosity of the liquid kg/m.s
- g – gravitational acceleration m/s2
- g0 — force conversion factor kgm/Ns2
- ρ1 — density of the liquid kg/m3
- ρv — density of vapour kg/m3
- σ — surface tension-liquid-vapour interface N/m
As can be seen, ΔT ∝ (q)⅓. This very important proportionality shows increasing ability of interface to transfer heat.
Nucleate Boiling – Flow Boiling
In flow boiling (or forced convection boiling), fluid flow is forced over a surface by external means such as a pump, as well as by buoyancy effects. Therefore, flow boiling is always accompanied by other convection effects. Conditions depend strongly on geometry, which may involve external flow over heated plates and cylinders or internal (duct) flow. In nuclear reactors, most of boiling regimes are just forced convection boiling. The flow boiling is also classified as either external and internal flow boiling depending on whether the fluid is forced to flow over a heated surface or inside a heated channel.
Internal flow boiling is much more complicated in nature than external flow boiling because there is no free surface for the vapor to escape, and thus both the liquid and the vapor are forced to flow together. The two-phase flow in a tube exhibits different flow boiling regimes, depending on the relative amounts of the liquid and the vapor phases. Therefore internal forced convection boiling is commonly referred to as two-phase flow.
Nucleate Boiling Correlations – Flow Boiling
In fully developed nucleate boiling with saturated coolant, the wall temperature is determined by local heat flux and pressure and is only slightly dependent on the Reynolds number. For subcooled water at absolute pressures between 0.1 – 0.6 MPa, McAdams correlation gives:
The Thom correlation is for the flow boiling (subcooled or saturated at pressures up to about 20 MPa) under conditions where the nucleate boiling contribution predominates over forced convection. This correlation is useful for rough estimation of expected temperature difference given the heat flux:
In 1963, Chen proposed the first flow boiling correlation for evaporation in vertical tubes to attain widespread use. Chen’s correlation includes both the heat transfer coefficients due to nucleate boiling as well as forced convective mechanisms. It must be noted, at higher vapor fractions, the heat transfer coefficient varies strongly with flow rate. The flow velocity in a core can be very high causing very high turbulences. This heat transfer mechanism has been referred to as “forced convection evaporation”. No adequate criteria has been established to determine the transition from nucleate boiling to forced convection vaporization. However, a single correlation that is valid for both nucleate boiling and forced convection vaporization has been developed by Chen for saturated boiling conditions and extended to include subcooled boiling by others. Chen proposed a correlation where the heat transfer coefficient is the sum of a forced convection component and a nucleate boiling component. It must be noted, the nucleate pool boiling correlation of Forster and Zuber (1955) is used to calculate the nucleate boiling heat transfer coefficient, hFZ and the turbulent flow correlation of Dittus-Boelter (1930) is used to calculate the liquid-phase convective heat transfer coefficient, hl.
The nucleate boiling suppression factor, S, is the ratio of the effective superheat to wall superheat. It accounts for decreased boiling heat transfer because the effective superheat across the boundary layer is less than the superheat based on wall temperature. The two-phase multiplier, F, is a function of the Martinelli parameter χtt.
Boiling Crisis – Critical Heat Flux
As was written, in nuclear reactors, limitations of the local heat flux is of the highest importance for reactor safety. For pressurized water reactors and also for boiling water reactors, there are thermal-hydraulic phenomena, which cause a sudden decrease in the efficiency of heat transfer (more precisely in the heat transfer coefficient). These phenomena occur at certain value of heat flux, known as the “critical heat flux”. The phenomena, that cause the deterioration of heat transfer are different for PWRs and for BWRs.
In both types of reactors, the problem is more or less associated with departure from nucleate boiling. The nucleate boiling heat flux cannot be increased indefinitely. At some value, we call it the “critical heat flux” (CHF), the steam produced can form an insulating layer over the surface, which in turn deteriorates the heat transfer coefficient. Immediately after the critical heat flux has been reached, boiling become unstable and film boiling occurs. The transition from nucleate boiling to film boiling is known as the “boiling crisis”. As was written, the phenomena, that cause the deterioration of heat transfer are different for PWRs and for BWRs.
Departure From Nucleate Boiling – DNB
In case of PWRs, the critical safety issue is named DNB (departure from nucleate boiling), which causes the formation of a local vapor layer, causing a dramatic reduction in heat transfer capability. This phenomenon occurs in the subcooled or low-quality region. The behaviour of the boiling crisis depends on many flow conditions (pressure, temperature, flow rate), but the boiling crisis occurs at a relatively high heat fluxes and appears to be associated with the cloud of bubbles, adjacent to the surface. These bubbles or film of vapor reduces the amount of incoming water. Since this phenomenon deteriorates the heat transfer coefficient and the heat flux remains, heat then accumulates in the fuel rod causing dramatic rise of cladding and fuel temperature. Simply, a very high temperature difference is required to transfer the critical heat flux being produced from the surface of the fuel rod to the reactor coolant (through vapor layer).
In case of PWRs, the critical flow is inverted annular flow, while in BWRs, the critical flow is usually annular flow. The difference in flow regime between post-dryout flow and post-DNB flow is depicted in the figure. In PWRs at normal operation the flow is considered to be single-phase. But a great deal of study has been performed on the nature of two-phase flow in case of transients and accidents (such as the loss-of-coolant accident – LOCA or trip of RCPs), which are of importance in reactor safety and in must be proved and declared in the Safety Analysis Report (SAR).
In pressurized water reactors, one of key safety requirements is that a departure from nucleate boiling (DNB) will not occur during steady state operation, normal operational transients, and anticipated operational occurrences (AOOs). Fuel cladding integrity will be maintained if the minimum DNBR remains above the 95/95 DNBR limit for PWRs ( a 95% probability at a 95% confidence level). DNB criterion is one of acceptance criteria in safety analyses as well as it constitutes one of safety limits in technical specifications.
An important duty of the plant operator is to control plant parameters such that a safe margin to DNB (or distance from DNB on the heat transfer curve) is maintained. Any sudden, large change in the following plant parameters/directions will decrease the margin to DNB:
- Decrease in reactor coolant pressure
- Decrease in reactor coolant flow rate
- Increase in reactor power
- Increase in reactor coolant inlet temperature
Therefore, the function of the operators and the plant design is to prevent a sudden, large change in these plant parameters.
Dryout – BWRs
In BWRs, similar phenomenon is known as “dryout” and it is directly associated with changes in flow pattern during evaporation in the high-quality region. At given combinations of flow rate through a channel, pressure, flow quality, and linear heat rate, the wall liquid film may exhaust and the wall may be dried out. At normal, the fuel surface is effectively cooled by boiling coolant. However when the heat flux exceeds a critical value (CHF – critical heat flux) the flow pattern may reach the dryout conditions (thin film of liquid disappears). The heat transfer from the fuel surface into the coolant is deteriorated, with the result of a drastically increased fuel surface temperature. In the high-quality region, the crisis occurs at a lower heat flux. Since the flow velocity in the vapor core is high, post-CHF heat transfer is much better than for low-quality critical flux (i.e. for PWRs temperature rises are higher and more rapid).